This study develops a concentration inequality for random variables that satisfy Bernstein’s condition when an appropriate cover is present. This inequality generalizes standard concentration results for common distributions, including Gamma, Exponential, and bounded random variables. We extend this analysis to financial risk management by deriving an upper bound for the Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) of dependent portfolios with an independent subgroup structure. Numerical simulations demonstrate that the derived upper bounds for the VaR and TVaR of a dependent portfolio are tight.
We study conditional distributions and their quantiles when a continuous covariate is conditioned on an interval event, without discretizing the covariate itself, focusing on bivariate copulas. For strict bivariate Archimedean copulas, we show that interval-conditioned quantiles on an arbitrary bin can be characterized by a one-dimensional equation involving only the copula generator. We identify settings in which numerical integration is unnecessary and provide closed-form expressions for interval-conditioned quantiles in the first (threshold) bin for the Clayton, Frank, Gumbel, and Joe copulas. For bivariate Clayton and Frank copulas, we further establish a unique anchor value inside the bin at which the interval-conditioned quantile equals the pointwise conditional quantile. We further show that last bin formulas can be obtained via the survival copula and derive closed-form last bin quantiles and anchors for the bivariate Frank copula. Numerical illustrations show that equal-probability bins can make distinct copula families appear nearly indistinguishable at central quantiles, whereas tail-favored bins restore separation. Our results offer practical formulas for interval-based inference, which is relevant to privacy-motivated binning.
Extending bivariate dependence concepts to higher dimensions is a challenging but essential task for a comprehensive understanding of multivariate dependence. Moreover, measuring overall dependence based on averages across the full domain of the joint distribution may fail to discern changes in dependence across different segments of the distribution, especially in the tails. In order to incorporate these features, we present the multivariate tail concentration function (TCF) as a graphical tool to assess both global and tail dependence. We show that this tool allows to represent multivariate dependence in a 2D plot regardless of the number of dimensions, it quantifies both lower and upper tail dependence at a finite scale, and it relates to multivariate Blomqvist’s beta. We propose to estimate the TCF non-parametrically using two methods and we compare their finite sample performance through a simulation study. To illustrate its practical application, we use the TCF to evaluate co-movements among the six Spanish banks included in the IBEX 35 stock index.
Bivariate time-to-event data often arise in various fields, including medicine, engineering, and economics, where understanding the association between two survival times is crucial. Traditional global association measures like Spearman’s rho and Kendall’s tau provide an average assessment, but fail to capture how association evolves over time. Local association measures, on the other hand, including the so-called cross ratio function (CRF), have been proposed to look at the association in more detail. This paper introduces a novel nonparametric estimator for the CRF applicable for univariate right-censored data, relying on Bernstein polynomials to obtain a smooth estimate of the bivariate survival copula, its partial derivatives, and the copula density. The proposed estimator’s finite-sample performance is evaluated through an elaborate simulation study and applied to real-life data, highlighting its practical utility and setting the stage for future research on local association in survival analysis.
We develop an amortized neural inference approach to assess the strength of tail dependence and the degree of asymmetry between the upper and lower tails based on a proposed unified tail dependence parameter for copulas. Extensive simulation studies are conducted for amortized inference with neural Bayes estimators of two full-range tail dependence copulas, to understand its performance under different situations, including training and inference speeds, comparisons of different methods for generating samples for training, comparisons between neural Bayes estimators and maximum likelihood estimators, performance for different sample sizes, performance for assessing tail dependence and tail asymmetry simultaneously, and modeling capacities in various misspecified situations. The proposed method has an ultrafast inference speed and is universally applicable and interpretable, making it useful for many real-world applications. An accompanying R package FastTail is also developed. To demonstrate its usefulness, we conducted an empirical study on stocks and ETFs from 2011 to 2025. The proposed GGEE-GARCH model using the neural Bayes estimators outperformed other commonly used copula GARCH models in predicting the next day Value-at-Risk. While the amortized neural inference approach is implemented for full-range tail dependence copulas, it can be useful for other parametric copulas with intractable likelihood functions, opening windows of opportunities for future development of new copula families with flexible dependence patterns.
This paper presents a displacement-like interpolation method between any two given checkerboard copulas. This interpolation is close to the geodesic connecting the two checkerboard copulas in the L 2-Wasserstein space, which, however, leaves the copula domain. In the proposed interpolation, mass is continuously shifted only across the boundaries of neighboring cells and copula property is ensured at each step. The method therefore can be interpreted as a discrete form of dynamical optimal transport constrained to the copula domain. We illustrate the interpolation with both a toy and a practical example, showing how the method yields dependence structures between two known states.
We study Gaussian-copula models with discrete margins, with primary emphasis on low-count (Poisson) data. Our goal is exact yet computationally efficient maximum likelihood (ML) estimation in regimes where many observations contain small counts, which imperils both identifiability and numerical stability. We develop three novel Kendall's tau-based approaches for initialization tailored to discrete margins in the low-count regime and embed it within an inference functions for margins (IFM) inspired start. We present three practical initializers (exact, low-intensity approximation, and a transformation-based approach) that substantially reduce the number of ML iterations and improve convergence. For the ML stage, we use an unconstrained reparameterization of the model's parameters using the log and spherical-Cholesky and compute exact rectangle probabilities. Analytical score functions are supplied throughout to stabilize Newton-type optimization. A simulation study across dimensions, dependence levels, and intensity regimes shows that the proposed initialization combined with exact ML achieves lower root-mean-squared error, lower bias and faster computation times than the alternative procedures. The methodology provides a pragmatic path to retain the statistical guarantees of ML (consistency, asymptotic normality, efficiency under correct specification) while remaining tractable for moderate- to high-dimensional discrete data. We conclude with guidance on initializer choice and discuss extensions to alternative correlation structures and different margins.
In recent years, a variety of novel measures of dependence have been introduced being capable of characterizing diverse types of directed dependence, hence diverse types of how a number of predictor variables X = (X 1, …, X p), p∈N $p\in \mathbb{N}$ , may affect a response variable Y. This includes perfect dependence of Y on X and independence between X and Y, but also less well-known concepts such as zero-explainability, stochastic comparability and complete separation. Certain such measures offer a representation in terms of the Markov product (Y, Y′), with Y′ being a conditionally independent copy of Y given X. This dimension reduction principle allows these measures to be estimated via the powerful nearest neighbor based estimation principle introduced in (Azadkia, M. and Chatterjee, S. (2021). A simple measure of conditional dependence. Ann. Stat. 49: 3070–3102). To achieve a deeper insight into the dimension reduction principle, this paper aims at translating the extreme variants of directed dependence, typically formulated in terms of the random vector (X, Y), into terms relating to its Markov product (Y, Y′).
Vine copulas, constructed using bivariate copulas as building blocks, provide a flexible framework for modeling multi-dimensional dependencies. However, this flexibility is accompanied by rapidly increasing complexity as dimensionality grows, necessitating appropriate truncation to manage this challenge. While use of Vuong's model selection test has been proposed as a method to determine the optimal truncation level, its application to vine copulas has been heuristic, assuming only strictly non-nested hypotheses. This assumption conflicts with the inherent nesting within truncated vine copula structures. In this paper, we systematically apply Vuong's model selection tests to distinguish competing models of truncated vine copulas under both nested and strictly non-nested hypotheses. Through extensive simulation studies, we characterize the conditions under which the nested hypotheses provide improved discernibility and demonstrate that the strictly non-nested framework can still yield valid distinctions in certain settings. This broader perspective on model comparison contributes to both methodological clarity and practical guidance for vine copula truncation.
This article introduces a hybrid framework that combines local Gaussian correlation (LGC) with hidden Markov models (HMMs) to model dynamic and nonlinear dependencies in general insurance claims, thereby addressing the limitations of static copula methods. When applied to Kenyan motor insurance claims (2008–2021) and Norwegian home insurance data (2012–2018), the proposed LGC-HMM approach captures regime-specific, nonlinear dependency patterns, revealing distinct stable and crisis periods through structural breaks in the dependency structure. Diagnostic checks confirm the HMM’s ability to reduce residual serial dependence, validating the latent state dynamics. Regime-aware value-at-risk (VaR) and tail VaR estimates derived from the LGC-HMM, using a proposed simulation procedure, outperform static copula models by adapting to structural changes, demonstrating robust forecasting performance. Visualization of forecasts via LGC maps further illustrates evolving tail dependencies. These findings support improved risk diversification and crisis-sensitive pricing strategies in actuarial practice.
This article investigates the effects of discrete marginal distributions on copula-based Markov chains. We establish results on mixing properties and parameter estimation for a copula-based Markov chain model with Bernoulli(pp) marginal distributions, emphasizing some distinctions between continuous and discrete state-space Markov chains. We derive parameter estimators using the maximum-likelihood estimation (MLE) method and explore alternative estimators of pp that are asymptotically equivalent to the MLE. Furthermore, we provide the asymptotic distributions of these parameter estimators. A simulation study is conducted to evaluate the performance of the various estimators for pp. Additionally, we employ the likelihood ratio test to assess independence within the sequence.
Due to their simple analytic form (bivariate) Archimedean copulas are usually viewed as very smooth and handy objects, which should distribute mass in a fairly regular and certainly not in a pathological way. Building upon recently established results on the Archimedean family and working with iterated function systems with probabilities, we falsify this natural conjecture and derive the surprising result that for every s∈s\hspace{0.33em}\in [1, 2] there exists some bivariate Archimedean copula As{A}_{s} fulfilling that the Hausdorff dimension of the support of As{A}_{s} is exactly ss.
This note is concerned with some historical remarks on and a partial review of two interesting mathematical subjects, the generalized Hoeffding-Fréchet functionals and the Monge-Kantorovich mass transportation problem. Both topics have a different motivation and history and are often considered in the literature as different subjects. The main aim of this review is to point out the close connection of these topics and to indicate some possibly fruitful relationships. For the class of Hoeffding-Fréchet functionals risk bounds for a lot of well-motivated additional model constraints have been worked out in recent years. These kinds of constraints are motivated by risk applications. We indicate some interesting connections of these developments to mass transportation problems as, e.g., to the solution of nonlinear mass transportation problems or to the use of stochastic ordering methods to the solution of mass transportation problems. We also briefly indicate the development of algorithms in the transportation problem by the regularization method (entropic optimal transport) and give corresponding references in the literature.
This article proposes a regression tree procedure to estimate conditional copulas. The associated algorithm determines classes of observations based on covariate values and fits a simple parametric copula model on each class. The association parameter changes from one class to another, allowing for non-linearity in the dependence structure modeling. It also allows the definition of classes of observations on which the so-called "simplifying assumption" holds reasonably well. When considering observations belonging to a given class separately, the association parameter no longer depends on the covariates according to our model. In this article, we derive asymptotic consistency results for the regression tree procedure and show that the proposed pruning methodology, i.e., the model selection techniques selecting the appropriate number of classes, is optimal in some sense. Simulations provide finite sample results, and an analysis of data of cases of human influenza presents the practical behavior of the procedure.
Kendall’s tau and conditional Kendall’s tau matrices are multivariate (conditional) dependence measures between the components of a random vector. For large dimensions, available estimators are computationally expensive and can be improved by averaging. Under structural assumptions on the underlying Kendall’s tau and conditional Kendall’s tau matrices, we introduce new estimators that have a significantly reduced computational cost while keeping a similar error level. In the unconditional setting, we assume that, up to reordering, the underlying Kendall’s tau matrix is block structured with constant values in each of the off-diagonal blocks. Consequences on the underlying correlation matrix are then discussed. The estimators take advantage of this block structure by averaging over (part of) the pairwise estimates in each of the off-diagonal blocks. Derived explicit variance expressions show their improved efficiency. In the conditional setting, the conditional Kendall’s tau matrix is assumed to have a block structure, for some value of the conditioning variable. Conditional Kendall’s tau matrix estimators are constructed similarly as in the unconditional case by averaging over (part of) the pairwise conditional Kendall’s tau estimators. We establish their joint asymptotic normality and show that the asymptotic variance is reduced compared to the naive estimators. Then, we perform a simulation study that displays the improved performance of both the unconditional and conditional estimators. Finally, the estimators are used for estimating the value at risk of a large stock portfolio; backtesting illustrates the obtained improvements compared to the previous estimators.
Comprehensive families of copulas including the three basic copulas (at least as limit cases) are useful tools to model countermonotonicity, independence, and comonotonicity of pairs of random variables on the same probability space. In this contribution, we study how the transition from a (basic) copula to a copula modeling a different dependence behavior can be realized by means of ordinal sums based on one of the three basic copulas, perturbing one of the three basic copulas (considering some appropriate parameterized transformations) and truncating the results using the Fr & eacute;chet-Hoeffding bounds. We provide results and examples showing the flexibility and the restrictions for obtaining new copulas or comprehensive families and illustrate the development of their dependence parameters.
This article studies the conditional dependency between random variables, conditionally upon a covariate (vector). The conditional copula fully characterizes this conditional dependency. A way to summarize this dependence structure taking into account the impact of the covariate is via the average conditional copula, which under fairly general conditions coincides with the partial copula. A mean is just one way to summarize this conditional dependence behaviour. In this article, we introduce the notions of median conditional copula and more generally quantile conditional copula. We investigate the existence of these concepts and establish explicit expressions for calculating them. Examples are given to illustrate the concepts, and the practical use of them is demonstrated in real data examples.
We analyze optimal low-rank approximations and correspondence analysis of the dependence structure given by arbitrary bivariate checkerboard copulas. Methodologically, we make use of the truncation of singular value decompositions of doubly stochastic matrices representing the copulas. The resulting (truncated) representations of the dependence structures are sparse, in particular, compared to the number of squares on the checkerboard. The additive structure of the decomposition carries through to statistical functionals of the copula, such as Kendall's tau \tau or Spearman's rho \rho , and also motivates similarity measures for checkerboard copulas. We link our analysis to continuous decompositions of copula densities and copula-generating algorithms and discuss further general properties of the decomposition and its truncation. For example, truncated series might lack nonnegativity, and approximation errors increase for monotonicity-like copulas. We provide algorithms and extensions that account for and counteract these properties. The low-rank representation is illustrated for various copula examples, and some analytical results are derived. The resulting correspondence analysis profile plots are analyzed, providing graphical insights into the dependence structure implied by the copula. An illustration is provided with an empirical data set on fuel injector spray characteristics in jet engines.
Motivated by recently investigated results on dependence measures and robust risk models, this article provides an overview of dependence properties of many well known bivariate copula families, where the focus is on the Schur order for conditional distributions, which has the fundamental property that minimal elements characterize independence and maximal elements characterize perfect directed dependence. We give conditions on copulas that imply the Schur ordering of the associated conditional distribution functions. For extreme-value copulas, we prove the equivalence of the lower orthant order, the Schur order for conditional distributions, and the pointwise order of the associated Pickands dependence functions. Furthermore, we provide several tables and figures that list and illustrate various positive dependence and monotonicity properties of copula families, in particular, from classes of Archimedean, extreme-value, and elliptical copulas. Finally, for Chatterjee’s rank correlation, which is consistent with the Schur order for conditional distributions, we give some new closed-form formulas in terms of the parameter of the underlying copula family.
